Flux surface: Difference between revisions

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:<math>\vec B \cdot \vec n = 0</math>
:<math>\vec B \cdot \vec n = 0</math>


everywhere. Defining a scalar ''flux function'' (''f'') such that its value is constant on the surface ''S'', this can be rewritten
everywhere on ''S''.  
In other words, the magnetic field does not ''cross'' the surface ''S'' anywhere, i.e., the magnetic flux traversing ''S'' is zero.
It is then possible to define a scalar ''flux function'' (''f'') such that its value is constant on the surface ''S'', and


:<math>\vec B \cdot \vec \nabla f = 0</math>
:<math>\vec B \cdot \vec \nabla f = 0</math>
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This fact lies at the basis of the design of magnetic confinement devices.
This fact lies at the basis of the design of magnetic confinement devices.


If a single vector field ''B'' has several such toroidal flux surfaces, they must necessarily be ''nested'' (since they cannot intersect) or be disjoint. Ignoring the latter possibility, it then makes sense to use the function ''f'' to label the flux surfaces, so ''f'' may be used as an effective "radial" coordinate. Each toroidal surface ''f'' encloses a volume ''V(f)''.
Assuming the flux surfaces have this toroidal topology, the function ''f'' defines a set of ''nested'' surfaces, so it makes sense to use this function to label the flux surfaces, i.e., ''f'' may be used as a "radial" coordinate. Each toroidal surface ''f'' encloses a volume ''V(f)''.
The surface corresponding to an infinitesimal volume ''V'' is essentially a line that corresponds to  
The surface corresponding to an infinitesimal volume ''V'' is essentially a line that corresponds to  
the ''toroidal axis'' (called ''magnetic axis'' when ''B'' is a magnetic field).
the ''toroidal axis'' (called ''magnetic axis'' when ''B'' is a magnetic field).
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:<math>F = \int_S{\vec B \cdot \vec n dS}</math>
:<math>F = \int_S{\vec B \cdot \vec n dS}</math>


[[File:Flux_definition.png|250px|thumb|right|Diagram showing the surfaces defining the poloidal (red) and toroidal (blue) flux]]
When ''B'' is a magnetic field with toroidal nested flux surfaces, two magnetic fluxes can be defined from two corresponding surfaces.
When ''B'' is a magnetic field with toroidal nested flux surfaces, two magnetic fluxes can be defined from two corresponding surfaces.
<ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) ISBN 0486432424</ref>
<ref>R.D. Hazeltine, J.D. Meiss, ''Plasma Confinement'', Courier Dover Publications (2003) {{ISBN|0486432424}}</ref>
The poloidal flux is defined by
The poloidal flux is defined by


:<math>\psi = \int_{S_p}{\vec B \cdot \vec n dS}</math>
:<math>\psi = \int_{S_p}{\vec B \cdot \vec n dS}</math>


where ''S<sub>p</sub>'' is a ring-shaped ribbon stretched between the magnetic axis and the flux surface ''f'', and the toroidal flux by
where ''S<sub>p</sub>'' is a ring-shaped ribbon stretched between the magnetic axis and the flux surface ''f''.
(Complementarily, ''S<sub>p</sub>'' can be taken to be a surface spanning the central hole of the torus.<ref>[http://link.aps.org/doi/10.1103/RevModPhys.76.1071 A.H. Boozer, ''Physics of magnetically confined plasmas'', Rev. Mod. Phys. '''76''' (2005) 1071 - 1141]</ref>)
Likewise, the toroidal flux is defined by


:<math>\phi = \int_{S_t}{\vec B \cdot \vec n dS}</math>
:<math>\phi = \int_{S_t}{\vec B \cdot \vec n dS}</math>
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It is natural to use &psi; or &phi; to label the flux surfaces instead of the unphysical label ''f''.
It is natural to use &psi; or &phi; to label the flux surfaces instead of the unphysical label ''f''.


== See also ==
* [[MHD equilibrium]]
* [[Toroidal coordinates]]
* [[Flux coordinates]]
* [[Rotational transform]]
* [[Magnetic shear]]
* [[Effective plasma radius]]
* [[Separatrix]]
* [[Flux tube]]


== References ==
== References ==
<references />
<references />